Abstract

Abstract: Based on a vector wave theory of volume holograms, dependence of holographic reconstruction on the polarization states of the writing and reading beams is discussed. It is found that under paraxial approximation the circular polarization holograms provide a better distinction of the reading beams. Characteristics of recording polarization holograms in thick phenanthrenequinone-doped poly(methyl methacrylate) (PQ/PMMA) photopolymer are experimentally investigated. It is found that the circular polarization holographic recording possesses better dynamic range and material sensitivity, and a uniform spatial frequency response over a wide range. The performance is comparable to that of the intensity holographic recording in PQ/PMMA. Based on theoretical analyses and the material properties, a polarization multiplexing holographic memory using circularly polarization recording configuration for increasing storage capacity has been designed and experimentally demonstrated.

© 2014 Optical Society of America

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References

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  1. G. Barbastathis and D. Psaltis, “Volume holographic multiplexing methods,” in Holographic data storage (Springer, 2000), pp. 21–62.
  2. T. Todorov, L. Nikolova, K. Stoyanova, and N. Tomova, “Polarization holography. 3: Some applications of polarization holographic recording,” Appl. Opt. 24(6), 785–788 (1985).
    [CrossRef] [PubMed]
  3. J. Eickmans, T. Bieringer, S. Kostromine, H. Berneth, and R. Thoma, “Photoaddressable polymers: a new class of materials for optical data storage and holographic memories,” Jpn. J. Appl. Phys. 38(3S), 1835–1836 (1999).
    [CrossRef]
  4. R. Hagen and T. Bieringer, “Photoaddressable polymers for optical data storage,” Adv. Mater. 13(23), 1805–1810 (2001).
    [CrossRef]
  5. L. L. Nedelchev, A. S. Matharu, S. Hvilsted, and P. S. Ramanujam, “Photoinduced anisotropy in a family of amorphous azobenzene polyesters for optical storage,” Appl. Opt. 42(29), 5918–5927 (2003).
    [CrossRef] [PubMed]
  6. H. Ono, T. Sekiguchi, A. Emoto, T. Shioda, and N. Kawatsuki, “Light wave propagation and Bragg diffraction in thick polarization gratings,” Jpn. J. Appl. Phys. 47(1010R), 7963–7967 (2008).
    [CrossRef]
  7. L. Nikolova and P. S. Ramanujam, Polarization holography (Cambridge University, 2009).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  12. D. Barada, Y. Kawagoe, H. Sekiguchi, T. Fukuda, S. Kawata, and T. Yatagai, “Volume polarization holography for optical data storage,” in SPIE OPTO, (International Society for Optics and Photonics, 2011), 79570Q.
  13. D. Barada, T. Ochiai, T. Fukuda, S. Kawata, K. Kuroda, and T. Yatagai, “Dual-channel polarization holography: a technique for recording two complex amplitude components of a vector wave,” Opt. Lett. 37(21), 4528–4530 (2012).
    [CrossRef] [PubMed]
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    [CrossRef]
  16. T. Huang and K. Wagner, “Coupled mode analysis of polarization volume hologram,” IEEE J. Quantum Electron. 31(2), 372–390 (1995).
    [CrossRef]
  17. L. Nikolova and T. Todorov, “Diffraction efficiency and selectivity of polarization holographic recording,” J. Mod. Opt. 31(5), 579–588 (1984).
  18. K. Y. Hsu, S. H. Lin, Y. N. Hsiao, and W. T. Whang, “Experimental characterization of phenanthrenequinone-doped poly(methyl methacrylate) photopolymer for volume holographic storage,” Opt. Eng. 42(5), 1390–1396 (2003).
    [CrossRef]
  19. S. H. Lin, P. L. Chen, and J. H. Lin, “Phenanthrenequinone-doped copolymers for holographic data storage,” Opt. Eng. 48(3), 035802-1- 035802-7 (2009).
  20. A. Trofimova, A. Stankevich, and V. Mogilnyi, “Phenanthrenequinone–polymethylmethacrylate composite for polarization phase recording,” J. Appl. Spectrosc. 76(4), 585–591 (2009).
    [CrossRef]
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    [CrossRef]
  22. S. H. Lin, P. L. Chen, C. I. Chuang, Y. F. Chao, and K. Y. Hsu, “Volume polarization holographic recording in thick phenanthrenequinone-doped poly(methyl methacrylate) photopolymer,” Opt. Lett. 36(16), 3039–3041 (2011).
    [CrossRef] [PubMed]

2012 (1)

2011 (2)

2010 (1)

C. I. Chuang, Y. N. Hsiao, S. H. Lin, and Y. F. Chao, “Real-time measurement of photo-induced effects in 9,10-phenanthrenequinone-doped poly(methyl methacrylate) photopolymer by phase-modulated ellipsometry,” Opt. Commun. 283(17), 3279–3283 (2010).
[CrossRef]

2009 (1)

A. Trofimova, A. Stankevich, and V. Mogilnyi, “Phenanthrenequinone–polymethylmethacrylate composite for polarization phase recording,” J. Appl. Spectrosc. 76(4), 585–591 (2009).
[CrossRef]

2008 (2)

H. Ono, T. Sekiguchi, A. Emoto, T. Shioda, and N. Kawatsuki, “Light wave propagation and Bragg diffraction in thick polarization gratings,” Jpn. J. Appl. Phys. 47(1010R), 7963–7967 (2008).
[CrossRef]

S. Pu, B. Yao, G. Liu, and Y. Wang, “Polarization multiplexing holographic optical recording of a new photochromic diarylethene,” Opt. Eng. 47(3), 030502 (2008).
[CrossRef]

2006 (1)

2004 (1)

2003 (2)

L. L. Nedelchev, A. S. Matharu, S. Hvilsted, and P. S. Ramanujam, “Photoinduced anisotropy in a family of amorphous azobenzene polyesters for optical storage,” Appl. Opt. 42(29), 5918–5927 (2003).
[CrossRef] [PubMed]

K. Y. Hsu, S. H. Lin, Y. N. Hsiao, and W. T. Whang, “Experimental characterization of phenanthrenequinone-doped poly(methyl methacrylate) photopolymer for volume holographic storage,” Opt. Eng. 42(5), 1390–1396 (2003).
[CrossRef]

2001 (1)

R. Hagen and T. Bieringer, “Photoaddressable polymers for optical data storage,” Adv. Mater. 13(23), 1805–1810 (2001).
[CrossRef]

1999 (2)

J. Eickmans, T. Bieringer, S. Kostromine, H. Berneth, and R. Thoma, “Photoaddressable polymers: a new class of materials for optical data storage and holographic memories,” Jpn. J. Appl. Phys. 38(3S), 1835–1836 (1999).
[CrossRef]

K. Kawano, T. Ishii, J. Minabe, T. Niitsu, Y. Nishikata, and K. Baba, “Holographic recording and retrieval of polarized light by use of polyester containing cyanoazobenzene units in the side chain,” Opt. Lett. 24(18), 1269–1271 (1999).
[CrossRef] [PubMed]

1995 (1)

T. Huang and K. Wagner, “Coupled mode analysis of polarization volume hologram,” IEEE J. Quantum Electron. 31(2), 372–390 (1995).
[CrossRef]

1985 (1)

1984 (1)

L. Nikolova and T. Todorov, “Diffraction efficiency and selectivity of polarization holographic recording,” J. Mod. Opt. 31(5), 579–588 (1984).

Baba, K.

Barada, D.

Berneth, H.

J. Eickmans, T. Bieringer, S. Kostromine, H. Berneth, and R. Thoma, “Photoaddressable polymers: a new class of materials for optical data storage and holographic memories,” Jpn. J. Appl. Phys. 38(3S), 1835–1836 (1999).
[CrossRef]

Bhattacharya, N.

Bieringer, T.

R. Hagen and T. Bieringer, “Photoaddressable polymers for optical data storage,” Adv. Mater. 13(23), 1805–1810 (2001).
[CrossRef]

J. Eickmans, T. Bieringer, S. Kostromine, H. Berneth, and R. Thoma, “Photoaddressable polymers: a new class of materials for optical data storage and holographic memories,” Jpn. J. Appl. Phys. 38(3S), 1835–1836 (1999).
[CrossRef]

Braat, J. J.

Cao, L.

Chan, V. S.

Chao, Y. F.

S. H. Lin, P. L. Chen, C. I. Chuang, Y. F. Chao, and K. Y. Hsu, “Volume polarization holographic recording in thick phenanthrenequinone-doped poly(methyl methacrylate) photopolymer,” Opt. Lett. 36(16), 3039–3041 (2011).
[CrossRef] [PubMed]

C. I. Chuang, Y. N. Hsiao, S. H. Lin, and Y. F. Chao, “Real-time measurement of photo-induced effects in 9,10-phenanthrenequinone-doped poly(methyl methacrylate) photopolymer by phase-modulated ellipsometry,” Opt. Commun. 283(17), 3279–3283 (2010).
[CrossRef]

Chen, P. L.

Chuang, C. I.

S. H. Lin, P. L. Chen, C. I. Chuang, Y. F. Chao, and K. Y. Hsu, “Volume polarization holographic recording in thick phenanthrenequinone-doped poly(methyl methacrylate) photopolymer,” Opt. Lett. 36(16), 3039–3041 (2011).
[CrossRef] [PubMed]

C. I. Chuang, Y. N. Hsiao, S. H. Lin, and Y. F. Chao, “Real-time measurement of photo-induced effects in 9,10-phenanthrenequinone-doped poly(methyl methacrylate) photopolymer by phase-modulated ellipsometry,” Opt. Commun. 283(17), 3279–3283 (2010).
[CrossRef]

Eickmans, J.

J. Eickmans, T. Bieringer, S. Kostromine, H. Berneth, and R. Thoma, “Photoaddressable polymers: a new class of materials for optical data storage and holographic memories,” Jpn. J. Appl. Phys. 38(3S), 1835–1836 (1999).
[CrossRef]

Emoto, A.

H. Ono, T. Sekiguchi, A. Emoto, T. Shioda, and N. Kawatsuki, “Light wave propagation and Bragg diffraction in thick polarization gratings,” Jpn. J. Appl. Phys. 47(1010R), 7963–7967 (2008).
[CrossRef]

Fujimura, R.

K. Kuroda, Y. Matsuhashi, R. Fujimura, and T. Shimura, “Theory of Polarization Holography,” Opt. Rev. 18(5), 374–382 (2011).
[CrossRef]

Fukuda, T.

Gu, C.

Hagen, R.

R. Hagen and T. Bieringer, “Photoaddressable polymers for optical data storage,” Adv. Mater. 13(23), 1805–1810 (2001).
[CrossRef]

He, Q.

Hsiao, Y. N.

C. I. Chuang, Y. N. Hsiao, S. H. Lin, and Y. F. Chao, “Real-time measurement of photo-induced effects in 9,10-phenanthrenequinone-doped poly(methyl methacrylate) photopolymer by phase-modulated ellipsometry,” Opt. Commun. 283(17), 3279–3283 (2010).
[CrossRef]

K. Y. Hsu, S. H. Lin, Y. N. Hsiao, and W. T. Whang, “Experimental characterization of phenanthrenequinone-doped poly(methyl methacrylate) photopolymer for volume holographic storage,” Opt. Eng. 42(5), 1390–1396 (2003).
[CrossRef]

Hsu, K. Y.

S. H. Lin, P. L. Chen, C. I. Chuang, Y. F. Chao, and K. Y. Hsu, “Volume polarization holographic recording in thick phenanthrenequinone-doped poly(methyl methacrylate) photopolymer,” Opt. Lett. 36(16), 3039–3041 (2011).
[CrossRef] [PubMed]

K. Y. Hsu, S. H. Lin, Y. N. Hsiao, and W. T. Whang, “Experimental characterization of phenanthrenequinone-doped poly(methyl methacrylate) photopolymer for volume holographic storage,” Opt. Eng. 42(5), 1390–1396 (2003).
[CrossRef]

Huang, T.

T. Huang and K. Wagner, “Coupled mode analysis of polarization volume hologram,” IEEE J. Quantum Electron. 31(2), 372–390 (1995).
[CrossRef]

Hvilsted, S.

Ishii, T.

Jin, G.

Kawano, K.

Kawata, S.

Kawatsuki, N.

H. Ono, T. Sekiguchi, A. Emoto, T. Shioda, and N. Kawatsuki, “Light wave propagation and Bragg diffraction in thick polarization gratings,” Jpn. J. Appl. Phys. 47(1010R), 7963–7967 (2008).
[CrossRef]

Koek, W. D.

Kostromine, S.

J. Eickmans, T. Bieringer, S. Kostromine, H. Berneth, and R. Thoma, “Photoaddressable polymers: a new class of materials for optical data storage and holographic memories,” Jpn. J. Appl. Phys. 38(3S), 1835–1836 (1999).
[CrossRef]

Kuroda, K.

Lin, S. H.

S. H. Lin, P. L. Chen, C. I. Chuang, Y. F. Chao, and K. Y. Hsu, “Volume polarization holographic recording in thick phenanthrenequinone-doped poly(methyl methacrylate) photopolymer,” Opt. Lett. 36(16), 3039–3041 (2011).
[CrossRef] [PubMed]

C. I. Chuang, Y. N. Hsiao, S. H. Lin, and Y. F. Chao, “Real-time measurement of photo-induced effects in 9,10-phenanthrenequinone-doped poly(methyl methacrylate) photopolymer by phase-modulated ellipsometry,” Opt. Commun. 283(17), 3279–3283 (2010).
[CrossRef]

K. Y. Hsu, S. H. Lin, Y. N. Hsiao, and W. T. Whang, “Experimental characterization of phenanthrenequinone-doped poly(methyl methacrylate) photopolymer for volume holographic storage,” Opt. Eng. 42(5), 1390–1396 (2003).
[CrossRef]

Liu, G.

S. Pu, B. Yao, G. Liu, and Y. Wang, “Polarization multiplexing holographic optical recording of a new photochromic diarylethene,” Opt. Eng. 47(3), 030502 (2008).
[CrossRef]

Matharu, A. S.

Matsuhashi, Y.

K. Kuroda, Y. Matsuhashi, R. Fujimura, and T. Shimura, “Theory of Polarization Holography,” Opt. Rev. 18(5), 374–382 (2011).
[CrossRef]

Minabe, J.

Mogilnyi, V.

A. Trofimova, A. Stankevich, and V. Mogilnyi, “Phenanthrenequinone–polymethylmethacrylate composite for polarization phase recording,” J. Appl. Spectrosc. 76(4), 585–591 (2009).
[CrossRef]

Nedelchev, L. L.

Niitsu, T.

Nikolova, L.

T. Todorov, L. Nikolova, K. Stoyanova, and N. Tomova, “Polarization holography. 3: Some applications of polarization holographic recording,” Appl. Opt. 24(6), 785–788 (1985).
[CrossRef] [PubMed]

L. Nikolova and T. Todorov, “Diffraction efficiency and selectivity of polarization holographic recording,” J. Mod. Opt. 31(5), 579–588 (1984).

Nishikata, Y.

Ochiai, T.

Ono, H.

H. Ono, T. Sekiguchi, A. Emoto, T. Shioda, and N. Kawatsuki, “Light wave propagation and Bragg diffraction in thick polarization gratings,” Jpn. J. Appl. Phys. 47(1010R), 7963–7967 (2008).
[CrossRef]

Pu, S.

S. Pu, B. Yao, G. Liu, and Y. Wang, “Polarization multiplexing holographic optical recording of a new photochromic diarylethene,” Opt. Eng. 47(3), 030502 (2008).
[CrossRef]

Ramanujam, P. S.

Sekiguchi, T.

H. Ono, T. Sekiguchi, A. Emoto, T. Shioda, and N. Kawatsuki, “Light wave propagation and Bragg diffraction in thick polarization gratings,” Jpn. J. Appl. Phys. 47(1010R), 7963–7967 (2008).
[CrossRef]

Shimura, T.

K. Kuroda, Y. Matsuhashi, R. Fujimura, and T. Shimura, “Theory of Polarization Holography,” Opt. Rev. 18(5), 374–382 (2011).
[CrossRef]

Shioda, T.

H. Ono, T. Sekiguchi, A. Emoto, T. Shioda, and N. Kawatsuki, “Light wave propagation and Bragg diffraction in thick polarization gratings,” Jpn. J. Appl. Phys. 47(1010R), 7963–7967 (2008).
[CrossRef]

Stankevich, A.

A. Trofimova, A. Stankevich, and V. Mogilnyi, “Phenanthrenequinone–polymethylmethacrylate composite for polarization phase recording,” J. Appl. Spectrosc. 76(4), 585–591 (2009).
[CrossRef]

Stoyanova, K.

Thoma, R.

J. Eickmans, T. Bieringer, S. Kostromine, H. Berneth, and R. Thoma, “Photoaddressable polymers: a new class of materials for optical data storage and holographic memories,” Jpn. J. Appl. Phys. 38(3S), 1835–1836 (1999).
[CrossRef]

Todorov, T.

T. Todorov, L. Nikolova, K. Stoyanova, and N. Tomova, “Polarization holography. 3: Some applications of polarization holographic recording,” Appl. Opt. 24(6), 785–788 (1985).
[CrossRef] [PubMed]

L. Nikolova and T. Todorov, “Diffraction efficiency and selectivity of polarization holographic recording,” J. Mod. Opt. 31(5), 579–588 (1984).

Tomova, N.

Trofimova, A.

A. Trofimova, A. Stankevich, and V. Mogilnyi, “Phenanthrenequinone–polymethylmethacrylate composite for polarization phase recording,” J. Appl. Spectrosc. 76(4), 585–591 (2009).
[CrossRef]

Wagner, K.

T. Huang and K. Wagner, “Coupled mode analysis of polarization volume hologram,” IEEE J. Quantum Electron. 31(2), 372–390 (1995).
[CrossRef]

Wang, Y.

S. Pu, B. Yao, G. Liu, and Y. Wang, “Polarization multiplexing holographic optical recording of a new photochromic diarylethene,” Opt. Eng. 47(3), 030502 (2008).
[CrossRef]

Wei, H.

Westerweel, J.

Whang, W. T.

K. Y. Hsu, S. H. Lin, Y. N. Hsiao, and W. T. Whang, “Experimental characterization of phenanthrenequinone-doped poly(methyl methacrylate) photopolymer for volume holographic storage,” Opt. Eng. 42(5), 1390–1396 (2003).
[CrossRef]

Xu, Z.

Yao, B.

S. Pu, B. Yao, G. Liu, and Y. Wang, “Polarization multiplexing holographic optical recording of a new photochromic diarylethene,” Opt. Eng. 47(3), 030502 (2008).
[CrossRef]

Yatagai, T.

Adv. Mater. (1)

R. Hagen and T. Bieringer, “Photoaddressable polymers for optical data storage,” Adv. Mater. 13(23), 1805–1810 (2001).
[CrossRef]

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

T. Huang and K. Wagner, “Coupled mode analysis of polarization volume hologram,” IEEE J. Quantum Electron. 31(2), 372–390 (1995).
[CrossRef]

J. Appl. Spectrosc. (1)

A. Trofimova, A. Stankevich, and V. Mogilnyi, “Phenanthrenequinone–polymethylmethacrylate composite for polarization phase recording,” J. Appl. Spectrosc. 76(4), 585–591 (2009).
[CrossRef]

J. Mod. Opt. (1)

L. Nikolova and T. Todorov, “Diffraction efficiency and selectivity of polarization holographic recording,” J. Mod. Opt. 31(5), 579–588 (1984).

Jpn. J. Appl. Phys. (2)

J. Eickmans, T. Bieringer, S. Kostromine, H. Berneth, and R. Thoma, “Photoaddressable polymers: a new class of materials for optical data storage and holographic memories,” Jpn. J. Appl. Phys. 38(3S), 1835–1836 (1999).
[CrossRef]

H. Ono, T. Sekiguchi, A. Emoto, T. Shioda, and N. Kawatsuki, “Light wave propagation and Bragg diffraction in thick polarization gratings,” Jpn. J. Appl. Phys. 47(1010R), 7963–7967 (2008).
[CrossRef]

Opt. Commun. (1)

C. I. Chuang, Y. N. Hsiao, S. H. Lin, and Y. F. Chao, “Real-time measurement of photo-induced effects in 9,10-phenanthrenequinone-doped poly(methyl methacrylate) photopolymer by phase-modulated ellipsometry,” Opt. Commun. 283(17), 3279–3283 (2010).
[CrossRef]

Opt. Eng. (2)

K. Y. Hsu, S. H. Lin, Y. N. Hsiao, and W. T. Whang, “Experimental characterization of phenanthrenequinone-doped poly(methyl methacrylate) photopolymer for volume holographic storage,” Opt. Eng. 42(5), 1390–1396 (2003).
[CrossRef]

S. Pu, B. Yao, G. Liu, and Y. Wang, “Polarization multiplexing holographic optical recording of a new photochromic diarylethene,” Opt. Eng. 47(3), 030502 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Opt. Rev. (1)

K. Kuroda, Y. Matsuhashi, R. Fujimura, and T. Shimura, “Theory of Polarization Holography,” Opt. Rev. 18(5), 374–382 (2011).
[CrossRef]

Other (5)

P. Yeh, Introduction to photorefractive nonlinear optics (Wiley, 1993).

L. Nikolova and P. S. Ramanujam, Polarization holography (Cambridge University, 2009).

G. Barbastathis and D. Psaltis, “Volume holographic multiplexing methods,” in Holographic data storage (Springer, 2000), pp. 21–62.

D. Barada, Y. Kawagoe, H. Sekiguchi, T. Fukuda, S. Kawata, and T. Yatagai, “Volume polarization holography for optical data storage,” in SPIE OPTO, (International Society for Optics and Photonics, 2011), 79570Q.

S. H. Lin, P. L. Chen, and J. H. Lin, “Phenanthrenequinone-doped copolymers for holographic data storage,” Opt. Eng. 48(3), 035802-1- 035802-7 (2009).

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of holographic recording (a) and reconstruction (b).

Fig. 2
Fig. 2

(a) Optical setup for recording and reconstructing polarization hologram in PQ/PMMA and polar coordinate plot for signal beam with (b) circular polarization and (c) linear polarization.

Fig. 3
Fig. 3

The cumulative grating strengths for orthogonal circular and linear polarizations as a function of exposure energy in the PQ/PMMA sample.

Fig. 4
Fig. 4

The cumulative grating strengths for different writing angles under orthogonally circular polarization recording configuration.

Fig. 5
Fig. 5

Optical setup for circular polarization multiplexing holographic memory experiment.

Fig. 6
Fig. 6

Optical experimental results. (a). the left photos are reconstructed from the index holograms and (b). the right photos are reconstructed from the polarization holograms. The number on the corner indicates the order of recording.

Tables (6)

Tables Icon

Table 1 Vector Amplitude F+ Reconstructed from Orthogonal Linear Polarization Hologram by Different Polarization States

Tables Icon

Table 2 Vector Amplitude F+ Reconstructed from Orthogonal Circular Polarization Hologram by Different Polarization States

Tables Icon

Table 3 Polarization States of the Diffracted Signals from the Intensity Hologram

Tables Icon

Table 4 Polarization States of the Diffracted Signals from a Polarization Hologram

Tables Icon

Table 5 Μ# and S for Different Recording Configurations with Orthogonally Polarized Writing Beams

Tables Icon

Table 6 M# and S For Different Writing Angles of Orthogonal Circular Polarization Recording Configuration

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

E= G + exp( i k S r )+ G - exp( i k R r )
k S =[ ksin θ S 0 kcos θ S ], k R =[ ksin θ R 0 kcos θ R ]
s ^ j =[ 0 1 0 ], p ^ j =[ cos θ j 0 sin θ j ]
n( r )= n 0 + n 1 cos[ ( k S k R ) r ]
n 1 G + G - *
F + ( G + G - * )[ F ( F k ^ S ) k ^ S ]
G + = G + L ^ S = G + 2 ( s ^ S +i p ^ S )= G + 2 [ icos θ S 1 isin θ S ], G = G L ^ R = G 2 ( s ^ R +i p ^ R )= G 2 [ icos θ R 1 isin θ R ] F = F L ^ R = G 2 ( s ^ R +i p ^ R )= G 2 [ icos θ R 1 isin θ R ]
F + G + G * F (1cos( θ S θ R )) 2 2 ( s ^ S +icos( θ S θ R ) p ^ S )
F + X + +{ X - ( X - k ^ S ) k ^ S }
X + = B 1 ( G - * F ) G + X =A( G + G - * ) F + B 2 ( G + F ) G - *
G + = G + s ^ S = G + [ 0 1 0 ], G = G p ^ R = G [ cos θ R 0 sin θ R ]
G + = G + p ^ S = G + [ cos θ S 0 sin θ S ], G = G s ^ R = G [ 0 1 0 ]
F - = F s ^ R = F [ 0 1 0 ], or F - = F p ^ R = F [ cos θ R 0 sin θ R ]
G + = G + L ^ S = G + 2 ( s ^ S +i p ^ S )= G + 2 [ icos θ S 1 isin θ S ], G = G R ^ R = G 2 ( s ^ R i p ^ R )= G 2 [ icos θ R 1 isin θ R ]
G + = G + R ^ S = G + 2 ( s ^ S i p ^ S )= G + 2 [ icos θ S 1 isin θ S ], G = G L ^ R = G 2 ( s ^ R +i p ^ R )= G 2 [ icos θ R 1 isin θ R ]
F = F L ^ R = F 2 ( s ^ R +i p ^ R )= F 2 [ icos θ R 1 isin θ R ], or F  = F R ^ R = F 2 ( s ^ R i p ^ R )= F 2 [ icos θ R 1 isin θ R ]
F + G * F G + [ A+ B 2 2 2 ]( 1 cos 2 ( θ S θ R ) )
F + G + G * F 2 2 ( 1cos( θ S θ R ) ) 2

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